I have the same issue. Which is how I found your question. I even replaced the bottom connector board and it still did not help. I guess the two small pins below the button have to hit the two pads on the buttom board. Where it says J3. But I tried shorting the J3 pads on my phone and that did not do anything.
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KobskiNov 29 '11 at 9:26

I replaced a screen that was actually for the 3G instead of 3GS and had this same issue. You may want to double check the new screen.
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Nate BirdDec 2 '11 at 2:31

But I didn't take off the back part, I just took off the screen and the frame, leaving the board with the phone back part as you can see in my screenshot that I haven't taken off the back part.
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MacMacDec 2 '11 at 13:10

check connector 4, if you touch that slightly and kncok it, the home will not work.
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Graeme HutchisonDec 2 '11 at 14:18

Hmm, I'm sure I didn't touch connector 4, I won't able to know if I touched it or not.
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MacMacDec 2 '11 at 15:17

its really easy to, so i would move that around until you start to get a response, as that is the connection for the home input button.
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Graeme HutchisonDec 2 '11 at 15:53

This is probably a bit of a long shot given the distinct possibility of this being a hardware rather than a software issues (because of historical circumstances). Having said that, I did have this same problem with my iPad 2 (was convinced that there was an issue with the hardware) and this was rectified by a 'hard device reset'.